Richard Statman

On the editing distance between unordered labeled trees

This paper considers the problem of computing the editing distance between unordered, labeled trees. We give efficient polynomial-time algorithms for the case when one tree is a string or has a bounded number of leaves. By contrast, we show that the problem is NP-complete even for binary trees having a label alphabet of size two.

Logical relations and the typed λ-calculus

Definition des relations logiques. Demonstration des principaux resultats syntaxiques sur le calcul de type lambda a partir du theoreme fondamental des relations logiques

Genus distributions for two classes of graphs

Abstract The set of orientable imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genus-respecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. It is proved that the genus distribution of any member of either class is strongly unimodal. These are the first two infinite classes of graphs for which such calculations have been achieved, except for a few classes, such as trees and cycles, whose members have all their cellular orientable imbeddings in the sphere.

Empty types in polymorphic lambda calculus

The model theory of simply typed and polymorphic (second-order) lambda calculus changes when types are allowed to be empty. For example, the “polymorphic Boolean” type really has <italic>exactly</italic> two elements in a polymorphic model only if the “absurd” type ∀<italic>t.t</italic> is empty. The standard β-ε axioms and equational inference rules which are complete when all types are nonempty are <italic>not complete</italic> for models with empty types. Without a little care about variable elimination, the standard rules are not even <italic>sound</italic> for empty types. We extend the standard system to obtain a complete proof system for models with empty types. The completeness proof is complicated by the fact that equational “term models” are not so easily obtained: in contrast to the nonempty case, not every theory with empty types is the theory of a single model.

On sets of solutions to combinator equations

Abstract We show that every r.e. set of combinators closed under β-conversion is the set of solutions to a pattern matching problem.

On the unification problem for Cartesian closed categories

An axiomatization of the isomorphisms that hold in all Cartesian closed categories (CCCs), discovered independently by S.V. Soloviev (1983) and by K.B. Bruce and G. Longo (1985), leads to seven equalities. It is shown that the unification problem for this theory is undecidable, thus setting an open question. It is also shown that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible. CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by M. Rittri (1990, 1991). It also has potential applications to the problem of polymorphic higher-order unification, which in turn is relevant to theorem proving, logic programming, and type reconstruction in higher-order languages.<<ETX>>

Some results on extensionality in lambda calculus

In this paper we consider the problem (due to A. Cantini) of the existence of a λ-theory T such that: –T is recursive enumerable; –the ω-rule holds in T (that is: if two terms M, N are such that for every closed term Q, MQ=NQ holds in T, then M=N holds in T). We solve affirmatively this problem. Some related questions are also discussed.

Morphisms and Partitions of V-sets

The set theoretic connection between functions and partitions is not worthy of further remark. Nevertheless, this connection turns out to have deep consequences for the theory of the Ershov numbering of lambda terms and thus for the connection between lambda calculus and classical recursion theory. Under the traditional understanding of lambda terms as function definitions, there are morphisms of the Ershov numbering of lambda terms which are not definable. This appears to be a serious incompleteness in the lambda calculus. However, we believe, instead, that this indefinability is a defect in our understanding of the functional nature of lambda terms. Below, for a different notion of lambda definition, we shall prove a representation theorem (completeness theorem) for morphisms. This theorem is based on a construction which realizes certain partitions as collections of fibers of morphisms defined by lambda terms in the classical sense of definition.

Normal Varieties of Combinators

We adopt for the most part the terminology and notation of [1]. A combinator is a term with no free variables. A set of combinators which is both recursively enumerable and closed under s conversion is said to be Visseral ([5]). Given combinators F and G, the variety defined by Fx = Gx is the set of all combinators M such that FM = GM. Such a variety is said to be normal if both F and G are normal. In this note we shall be principally concerned with normal varieties.

The word problem for Smullyan's lark combinator is decidable

We show that the problem of deciding whether two applicative combinations of Smullyans L combinator are equal is solvable.